My 2019 Mathematics A To Z: Hamiltonian

Today’s A To Z term is another I drew from Mr Wu, of the Singapore Math Tuition blog. It gives me more chances to discuss differential equations and mathematical physics, too.

The Hamiltonian we name for Sir William Rowan Hamilton, the 19th century Irish mathematical physicists who worked on everything. You might have encountered his name from hearing about quaternions. Or for coining the terms “scalar” and “tensor”. Or for work in graph theory. There’s more. He did work in Fourier analysis, which is what you get into when you feel at ease with Fourier series. And then wild stuff combining matrices and rings. He’s not quite one of those people where there’s a Hamilton’s Theorem for every field of mathematics you might be interested in. It’s close, though.

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Hamiltonian.

When you first learn about physics you learn about forces and accelerations and stuff. When you major in physics you learn to avoid dealing with forces and accelerations and stuff. It’s not explicit. But you get trained to look, so far as possible, away from vectors. Look to scalars. Look to single numbers that somehow encode your problem.

A great example of this is the Lagrangian. It’s built on “generalized coordinates”, which are not necessarily, like, position and velocity and all. They include the things that describe your system. This can be positions. It’s often angles. The Lagrangian shines in problems where it matters that something rotates. Or if you need to work with polar coordinates or spherical coordinates or anything non-rectangular. The Lagrangian is, in your general coordinates, equal to the kinetic energy minus the potential energy. It’ll be a function. It’ll depend on your coordinates and on the derivative-with-respect-to-time of your coordinates. You can take partial derivatives of the Lagrangian. This tells how the coordinates, and the change-in-time of your coordinates should change over time.

The Hamiltonian is a similar way of working out mechanics problems. The Hamiltonian function isn’t anything so primitive as the kinetic energy minus the potential energy. No, the Hamiltonian is the kinetic energy plus the potential energy. Totally different in idea.

From that description you maybe guessed you can transfer from the Lagrangian to the Hamiltonian. Maybe vice-versa. Yes, you can, although we use the term “transform”. Specifically a “Legendre transform”. We can use any coordinates we like, just as with Lagrangian mechanics. And, as with the Lagrangian, we can find how coordinates change over time. The change of any coordinate depends on the partial derivative of the Hamiltonian with respect to a particular other coordinate. This other coordinate is its “conjugate”. (It may either be this derivative, or minus one times this derivative. By the time you’re doing work in the field you’ll know which.)

That conjugate coordinate is the important thing. It’s why we muck around with Hamiltonians when Lagrangians are so similar. In ordinary, common coordinate systems these conjugate coordinates form nice pairs. In Cartesian coordinates, the conjugate to a particle’s position is its momentum, and vice-versa. In polar coordinates, the conjugate to the angular velocity is the angular momentum. These are nice-sounding pairs. But that’s our good luck. These happen to match stuff we already think is important. In general coordinates one or more of a pair can be some fusion of variables we don’t have a word for and would never care about. Sometimes it gets weird. In the problem of vortices swirling around each other on an infinitely great plane? The horizontal position is conjugate to the vertical position. Velocity doesn’t enter into it. For vortices on the sphere the longitude is conjugate to the cosine of the latitude.

What’s valuable about these pairings is that they make a “symplectic manifold”. A manifold is a patch of space where stuff works like normal Euclidean geometry does. In this case, the space is in “phase space”. This is the collection of all the possible combinations of all the variables that could ever turn up. Every particular moment of a mechanical system matches some point in phase space. Its evolution over time traces out a path in that space. Call it a trajectory or an orbit as you like.

We get good things from looking at the geometry that this symplectic manifold implies. For example, if we know that one variable doesn’t appear in the Hamiltonian, then its conjugate’s value never changes. This is almost the kindest thing you can do for a mathematical physicist. But more. A famous theorem by Emmy Noether tells us that symmetries in the Hamiltonian match with conservation laws in the physics. Time-invariance, for example — time not appearing in the Hamiltonian — gives us the conservation of energy. If only distances between things, not absolute positions, matter, then we get conservation of linear momentum. Stuff like that. To find conservation laws in physics problems is the kindest thing you can do for a mathematical physicist.

The Hamiltonian was born out of planetary physics. These are problems easy to understand and, apart from the case of one star with one planet orbiting each other, impossible to solve exactly. That’s all right. The formalism applies to all kinds of problems. They’re very good at handling particles that interact with each other and maybe some potential energy. This is a lot of stuff.

More, the approach extends naturally to quantum mechanics. It takes some damage along the way. We can’t talk about “the” position or “the” momentum of anything quantum-mechanical. But what we get when we look at quantum mechanics looks very much like what Hamiltonians do. We can calculate things which are quantum quite well by using these tools. This though they came from questions like why Saturn’s rings haven’t fallen part and whether the Earth will stay around its present orbit.

It holds surprising power, too. Notice that the Hamiltonian is the kinetic energy of a system plus its potential energy. For a lot of physics problems that’s all the energy there is. That is, the value of the Hamiltonian for some set of coordinates is the total energy of the system at that time. And, if there’s no energy lost to friction or heat or whatever? Then that’s the total energy of the system for all time.

Here’s where this becomes almost practical. We often want to do a numerical simulation of a physics problem. Generically, we do this by looking up what all the values of all the coordinates are at some starting time t₀. Then we calculate how fast these coordinates are changing with time. We pick a small change in time, Δ t. Then we say that at time t₀ plus Δ t, the coordinates are whatever they started at plus Δ t times that rate of change. And then we repeat, figuring out how fast the coordinates are changing now, at this position and time.

The trouble is we always make some mistake, and once we’ve made a mistake, we’re going to keep on making mistakes. We can do some clever stuff to make the smallest error possible figuring out where to go, but it’ll still happen. Usually, we stick to calculations where the error won’t mess up our results.

But when we look at stuff like whether the Earth will stay around its present orbit? We can’t make each step good enough for that. Unless we get to thinking about the Hamiltonian, and our symplectic variables. The actual system traces out a path in phase space. Everyone on that path the Hamiltonian is a particular value, the energy of the system. So use the regular methods to project most of the variables to the new time, t₀ + Δ t. But the rest? Pick the values that make the Hamiltonian work out right. Also momentum and angular momentum and other stuff we know get conserved. We’ll still make an error. But it’s a different kind of error. It’ll project to a point that’s maybe in the wrong place on the trajectory. But it’s on the trajectory.

(OK, it’s near the trajectory. Suppose the real energy is, oh, the square root of 5. The computer simulation will have an energy of 2.23607. This is close but not exactly the same. That’s all right. Each step will stay close to the real energy.)

So what we’ll get is a projection of the Earth’s orbit that maybe puts it in the wrong place in its orbit. Putting the planet on the opposite side of the sun from Venus when we ought to see Venus transiting the Sun. That’s all right, if what we’re interested in is whether Venus and Earth are still in the solar system.

There’s a special cost for this. If there weren’t we’d use it all the time. The cost is computational complexity. It’s pricey enough that you haven’t heard about these “symplectic integrators” before. That’s all right. These are the kinds of things open to us once we look long into the Hamiltonian.

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This wraps up my big essay-writing for the week. I will pluck some older essays out of obscurity to re-share tomorrow and Saturday. All of Fall 2019 A To Z posts should be at this link. Next week should have the letter I on Tuesday and J on Thursday. All of my A To Z essays should be available at this link. And I am still interested in topics I might use for the letters K through N. Thank you.

What’s The Point Of Hamiltonian Mechanics?

The Diff_Eq twitter feed had a link the other day to a fine question put on StackExchange.com: What’s the Point of Hamiltonian Mechanics? Hamiltonian Mechanics is a different way of expressing the laws of Newtonian mechanics from what you learn in high school, and what you learn from that F equals m a business, and it gets introduced in the Mechanics course you take early on as a physics major.

At this level of physics you’re mostly concerned with, well, the motion of falling balls, of masses hung on springs, of pendulums swinging back and forth, of satellites orbiting planets. This is all nice tangible stuff and you can work problems out pretty well if you know all the forces the moving things exert on one another, forming a lot of equations that tell you how the particles are accelerating, from which you can get how the velocities are changing, from which you can get how the positions are changing.

The Hamiltonian formation starts out looking like it’s making life harder, because instead of looking just at the positions of particles, it looks at both the positions and the momentums (which is the product of the mass and the velocity). However, instead of looking at the forces, particularly, you look at the energy in the system, which typically is going to be the kinetic energy plus the potential energy. The energy is a nice thing to look at, because it’s got some obvious physical meaning, and you should know how it changes over time, and because it’s just a number (a scalar, in the trade) instead of a vector, the way forces are.

And here’s a neat thing: the way the position changes over time is found by looking at how the energy would change if you made a little change in the momentum; and the way the momentum changes over time is found by looking at how the energy would change if you made a little change in the position. As that sentence suggests, that’s awfully pretty; there’s something aesthetically compelling about treating position and momentum so very similarly. (They’re not treated in exactly the same way, but it’s close enough.) And writing the mechanics problem this way, as position and momentum changing in time, means we can use tools that come from linear algebra and the study of matrices to answer big questions like whether the way the system moves is stable, which are hard to answer otherwise.

The questioner who started the StackExchange discussion pointed out that before they get to Hamiltonian mechanics, the course also introduced the Euler-Lagrange formation, which looks a lot like the Hamiltonian, and which was developed first, and gets introduced to students first; why not use that? Here I have to side with most of the commenters about the Hamiltonian turning out to be more useful when you go on to more advanced physics. The Euler-Lagrange form is neat, and particularly liberating because you get an incredible freedom in how you set up the coordinates describing the action of your system. But it doesn’t have that same symmetry in treating the position and momentum, and you don’t get the energy of the system built right into the equations you’re writing, and you can’t use the linear algebra and matrix tools that were introduced. Mostly, the good things that Euler-Lagrange forms give you, such as making it obvious when a particular coordinate doesn’t actually contribute to the behavior of the system, or letting you look at energy instead of forces, the Hamiltonian also gives you, and the Hamiltonian can be used to do more later on.